On cyclotomic matrices related to Kloosterman sums over finite fields

Abstract: Let $p$ be an odd prime and $\mathbb{F}p$ be the finite field with $p$ elements. For any $a,b\in\mathbb{F}_p$, it is known that the Kloosterman sum $$K_p(a,b)=\sum{x\in\mathbb{F}p\setminus{0}}e^{\frac{2πi}{p}(ax+\frac{b}{x})}$$ can be viewed as a finite field analogue of certain Bessel function. In this paper, using the arithmetic properties of character sums over $\mathbb{F}_p$, we study some cyclotomic matrices involving Kloosterman sums. For example, we prove that the matrix $[K_p(1,i^2+j^2)]{1\le i,j\le (p-1)/2}$ is singular if and only if $p\ge11$.